Rosenbrock system matrix

The Rosenbrock System Matrix (or Rosenbrock's system matrix) of a linear time invariant system is a useful representation bridging state-space representation and transfer function matrix form. It was proposed in 1967 by Howard H. Rosenbrock.[1]

Definition

Consider the dynamic system

The Rosenbrock system matrix is given by

In the original work by Rosenbrock, the constant matrix is allowed to be a polynomial in .

The transfer function between the input and output is given by

where is the column of and is the row of .

Based in this representation, Rosenbrock developed his version of the PHB test.

Short form

For computational purposes, a short form of the Rosenbrock system matrix is more appropriate[2] and given by

The short form of the Rosenbrock system matrix has been widely used in H-infinity methods in control theory, where it is also referred to as packed form; see command pck in.[3] An interpretation of the Rosenbrock System Matrix as a Linear Fractional Transformation can be found in.[4]

One of the first applications of the Rosenbrock form was the development of an efficient computational method for Kalman decomposition, which is based on the pivot element method. A variant of Rosenbrock’s method is implemented in the minreal command of Matlab.[5] as well as GNU Octave.

References

  1. Rosenbrock, H.H. (1967). "Transformation of linear constant system equations". Proc. I.E.E. 114: 541–544.
  2. Rosenbrock, H. H. (1970). State-Space and Multivariable Theory. Nelson.
  3. "Mu Analysis and Synthesis Toolbox". Retrieved 25 August 2014.
  4. Zhou, Kemin; Doyle, John C.; Glover, Keith (1995). Robust and Optimal Control. Prentice Hall.
  5. De Schutter, B. (2000). "Minimal state-space realization in linear system theory: an overview". Journal of Computational and Applied Mathematics. 121 (1-2): 331–354. doi:10.1016/S0377-0427(00)00341-1.
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